An Unconditional BGST\(\toDMATH0R_2(T; w) \;=\; \frac{T\log T}{2\pi}\!\int_{\mathbb{R}}\!\Bigl(1 - \bigl(\tfrac{\sin\pi v}{\pi v}\bigr)^2\Bigr)\, w(v)\, dv \;+\; O\!\bigl(T\sqrt{\log T}\,\|\widehat{w}\|_{L^1}\bigr).DMATH1N(T) \;=\; \#\{\gamma : 0 < \gamma \le T\} \;=\; \frac{T}{2\pi}\log\frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T).DMATH2F(\alpha, T) \;:=\; \frac{2\pi}{TL}\,\sum_{0<\gamma,\gamma'\le T} T^{i\alpha(\gamma-\gamma')}\cdot \frac{4}{4+(\gamma-\gamma')^2}.DMATH3F(\alpha, T) \;=\; T^{-2\alpha}(L + O(1)) + \alpha + o(1) \qquad (\text{RH}).DMATH4F(\alpha, T) \;=\; T^{-2\alpha}\bigl(L + O(1)\bigr) + \alpha + O\!\bigl(1/\sqrt{L}\bigr).DMATH5R_2(T; w) \;:=\; \sum_{\substack{0<\gamma,\gamma'\le T\\ \gamma\ne\gamma'}} w\!\Bigl(\frac{(\gamma-\gamma')L}{2\pi}\Bigr), \tag{1.1}DMATH6R_2(T; w) \;=\; \frac{TL}{2\pi}\bigl[\widehat{w}(0) + A_w - w(0)\bigr] \;+\; O\!\bigl(T\sqrt{L}\,\|\widehat{w}\|_{L^1[-1,1]}\bigr), \tag{1.2}DMATH7R_2(T; w) \;=\; \frac{TL}{2\pi}\!\int_{\mathbb{R}}\!\Bigl(1 - \Bigl(\frac{\sin\pi v}{\pi v}\Bigr)^2\Bigr)\, w(v)\,dv \;+\; O\!\bigl(T\sqrt{L}\,\|\widehat{w}\|_{L^1}\bigr). \tag{1.3}DMATH8w(v) \;=\; \int_{\mathbb{R}} \widehat{w}(\alpha)\,e^{2\pi i v\alpha}\,d\alpha \;=\; \int_{-1}^{1}\widehat{w}(\alpha)\,e^{2\pi i v\alpha}\,d\alpha,DMATH9w\!\Bigl(\frac{(\gamma-\gamma')L}{2\pi}\Bigr) \;=\; \int_{-1}^{1}\widehat{w}(\alpha)\,T^{i\alpha(\gamma-\gamma')}\,d\alpha.DMATH10R_2(T; w) \;=\; \int_{-1}^{1}\widehat{w}(\alpha)\,G(\alpha, T)\,d\alpha,DMATH11G(\alpha, T) \;:=\; \sum_{\substack{0<\gamma,\gamma'\le T\\ \gamma\ne\gamma'}} T^{i\alpha(\gamma-\gamma')}. \tag{3.1}DMATH12G_0(\alpha, T) \;:=\; \sum_{0<\gamma,\gamma'\le T} T^{i\alpha(\gamma-\gamma')} \;=\; N(T) + G(\alpha, T),DMATH13\boxed{\; R_2(T; w) \;=\; \int_{-1}^{1}\widehat{w}(\alpha)\,G_0(\alpha, T)\,d\alpha \;-\; N(T)\,w(0). \;} \tag{3.2}DMATH14\widehat{w_M}(\xi) \;=\; \int_{\mathbb{R}}\frac{4}{4+u^2}\,e^{-2\pi i u\xi}\,du \;=\; 2\pi\,e^{-4\pi|\xi|}.DMATH15\frac{4}{4+u^2} \;=\; \int_{\mathbb{R}} 2\pi\,e^{-4\pi|\xi|}\,e^{2\pi i u\xi}\,d\xi. \tag{4.1}DMATH16\frac{TL}{2\pi}\,F(\alpha, T) \;=\; \sum_{\gamma,\gamma'} T^{i\alpha(\gamma-\gamma')}\int_{\mathbb{R}} 2\pi\,e^{-4\pi|\xi|}\,e^{i(\gamma-\gamma')\cdot 2\pi\xi}\,d\xi \;=\; \int 2\pi\,e^{-4\pi|\xi|}\,G_0\!\bigl(\alpha + \tfrac{2\pi\xi}{L},\,T\bigr)\,d\xi.DMATH17\frac{TL}{2\pi}\,F(\alpha, T) \;=\; L\int_{\mathbb{R}} e^{-2L|\beta-\alpha|}\,G_0(\beta, T)\,d\beta,DMATH18\boxed{\; \frac{TL}{2\pi}\,F(\alpha, T) \;=\; (\mathcal{K}_L \ast G_0)(\alpha, T), \qquad \mathcal{K}_L(u) = Le^{-2L|u|}. \;} \tag{4.2}DMATH19\widehat{\mathcal{K}_L}(\xi) \;=\; \frac{L^2}{L^2 + \pi^2\xi^2},DMATH20\widetilde{w}(\alpha) \;:=\; (\mathcal{K}_L^{-1} \ast \widehat{w})(\alpha). \tag{5.1}DMATH21\widetilde{w}(\alpha) \;=\; \widehat{w}(\alpha) \;-\; \frac{1}{4L^2}\,\widehat{w}''(\alpha). \tag{5.2}DMATH22\|\widetilde{w} - \widehat{w}\|_{L^1(\mathbb{R})} \;\le\; \frac{1}{4L^2}\|\widehat{w}''\|_{L^1([-1,1])} \;=\; O(\|\widehat{w}\|_{C^2}/L^2). \tag{5.3}DMATH23\int_{\mathbb{R}}\widehat{w}(\alpha)\,G_0(\alpha, T)\,d\alpha \;=\; \frac{TL}{2\pi}\int_{\mathbb{R}}\widetilde{w}(\alpha)\,F(\alpha, T)\,d\alpha. \tag{5.4}DMATH24R_2(T; w) \;=\; \frac{TL}{2\pi}\int_{\mathbb{R}}\widetilde{w}(\alpha)\,F(\alpha, T)\,d\alpha \;-\; N(T)\,w(0). \tag{5.5}DMATH25F(\alpha, T) \;=\; T^{-2|\alpha|}\bigl(L + O(1)\bigr) + |\alpha| + O(1/\sqrt{L}).DMATH26\frac{TL}{2\pi}\int_{-1}^{1}\widetilde{w}(\alpha)\,F(\alpha, T)\,d\alpha \;=\; M_1 + M_2 + E_1 + E_2,DMATH27M_1 \;=\; \frac{TL^2}{2\pi}\int_{-1}^{1}\widetilde{w}(\alpha)\,T^{-2|\alpha|}\,d\alpha.DMATH28\int_{-1}^{1}\widetilde{w}(\alpha)\,e^{-2L|\alpha|}\,d\alpha \;=\; \widetilde{w}(0)\cdot\!\int_{-\infty}^{\infty}\!e^{-2L|\alpha|}\,d\alpha \;+\; O(\|\widetilde{w}'\|_\infty/L^2) \;=\; \frac{\widetilde{w}(0)}{L} + O(1/L^2),DMATH29M_1 \;=\; \frac{TL}{2\pi}\,\widehat{w}(0)\,\bigl(1 + O(1/L)\bigr). \tag{6.1}DMATH30M_2 \;=\; \frac{TL}{2\pi}\int_{-1}^{1}\widetilde{w}(\alpha)\,|\alpha|\,d\alpha \;=\; \frac{TL}{2\pi}\bigl(A_w + O(\|\widehat{w}\|_{C^2}/L^2)\bigr), \tag{6.2}DMATH31E_1 \;=\; \frac{TL}{2\pi}\int_{-1}^{1}\widetilde{w}(\alpha)\,T^{-2|\alpha|}\,O(1)\,d\alpha \;=\; O(T\|\widehat{w}\|_\infty) \;=\; O(T). \tag{6.3}DMATH32E_2 \;=\; \frac{TL}{2\pi}\int_{-1}^{1}\widetilde{w}(\alpha)\cdot O(1/\sqrt{L})\,d\alpha \;=\; O\!\Bigl(\frac{TL\,\|\widetilde{w}\|_{L^1[-1,1]}}{\sqrt{L}}\Bigr).DMATH33E_2 \;=\; O\!\bigl(T\sqrt{L}\,\|\widehat{w}\|_{L^1[-1,1]}\bigr). \tag{6.4}DMATH34\frac{TL}{2\pi}\int_{|\alpha|>1}\widetilde{w}(\alpha)\,F(\alpha,T)\,d\alpha \;=\; 0. \tag{6.5}DMATH35R_2(T; w) \;=\; \frac{TL}{2\pi}\widehat{w}(0) \;+\; \frac{TL}{2\pi} A_w \;+\; O(T) \;+\; O(T\sqrt{L}\,\|\widehat{w}\|_{L^1}) \;-\; N(T)\,w(0). \tag{7.1}DMATH36R_2(T; w) \;=\; \frac{TL}{2\pi}\bigl[\widehat{w}(0) + A_w - w(0)\bigr] + O\!\bigl(T\sqrt{L}\,\|\widehat{w}\|_{L^1}\bigr). \tag{7.2}DMATH37\int_{-1}^{1}(1 - |\alpha|)\widehat{w}(\alpha)\,d\alpha \;=\; \int_{\mathbb{R}}\!\Bigl(\frac{\sin\pi v}{\pi v}\Bigr)^{\!2}\, w(v)\,dv.DMATH38\widehat{w}(0) + A_w - w(0) \;=\; \widehat{w}(0) - \int_{-1}^{1}(1-|\alpha|)\widehat{w}(\alpha)\,d\alpha \;=\; \int_{\mathbb{R}}\Bigl(1 - \Bigl(\frac{\sin\pi v}{\pi v}\Bigr)^{\!2}\Bigr)\,w(v)\,dv,DMATH39R_2(T; w) \;=\; \frac{TL}{2\pi}\int_{\mathbb{R}}\Bigl(1 - \Bigl(\frac{\sin\pi v}{\pi v}\Bigr)^{\!2}\Bigr)\, w(v)\,dv \;+\; O\!\bigl(T\sqrt{\log T}\,\|\widehat{w}\|_{L^1}\bigr). \qquad\squareDMATH40R_2(T;w) = \int_{-1}^{1}\widehat{w}(\alpha)\,|S(\alpha,T)|^2\,d\alpha \;-\; N(T)\,w(0), \qquad S(\alpha,T) := \sum_{0<\gamma\le T} T^{i\alpha\gamma},DMATH41P_c(T) \;:=\; \#\!\bigl\{(\gamma,\gamma') : 0 < \gamma \ne \gamma' \le T,\; |\gamma - \gamma'| \le c\cdot\tfrac{2\pi}{L}\bigr\}.DMATH42P_c(T) \;=\; \frac{TL}{2\pi}\!\int_{-c}^{c}\!\bigl(1 - \text{sinc}^2(v)\bigr)\,dv \;+\; O\bigl(T\sqrt{L}\bigr) \qquad (T \to \infty),DMATH43w_{-}(v) \;\le\; \mathbf{1}_{[-c,c]}(v) \;\le\; w_{+}(v), \qquad \int_{\mathbb{R}} (w_{\pm}(v) - \mathbf{1}_{[-c,c]}(v))\,dv \;\le\; \epsilon.DMATH44P_c(T) \;\le\; R_2(T; w_+) \;=\; \frac{TL}{2\pi}\!\int_{\mathbb{R}}\!\bigl(1 - \text{sinc}^2(v)\bigr)\,w_+(v)\,dv \;+\; O(T\sqrt{L}).DMATH45\gamma_{n+1} - \gamma_n \;\le\; (1+\epsilon)\,\frac{2\pi}{L}\)$

for any fixed \(\epsilon > 0\), where \(c = c(\epsilon) > 0\) is an absolute constant.

This follows from Proposition 1 with \(c = 1+\epsilon\): the density \(\int_0^{1+\epsilon}2g_2 > 0\) forces a positive proportion of gaps to be small. Under RH, this is a consequence of Montgomery (1973); our transfer makes it unconditional.

11.3. Remark: variance of the zero-counting function

The pair-correlation formula controls the variance of \(N(t+H) - N(t)\) for short intervals \(H = O(1/L)\), via a classical reduction to \(R_2(T; w_h)\) for a suitable bandlimited test function (see [IK04, §15.4] and [Mon73, §4]). The Main Theorem makes this connection unconditional.

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Acknowledgments

This note is an extract from a larger investigation into the mesoscopic structure of zero-pair correlations (the de Branges–Szegő chain program). The present transfer result is classical Fourier analysis; routine verification of the Parseval, convolution, and Fourier-inversion identities used in §3–§5 was performed with automated proof-checking tools.

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References

[BGST23] S. Baluyot, D. A. Goldston, A. I. Suriajaya, C. L. Turnage-Butterbaugh, An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta function, Acta Arithmetica 214 (2024), 357–376; arXiv:2306.04799.

[GGÖS00] D. A. Goldston, S. M. Gonek, A. E. Özlük, C. Snyder, On the pair correlation of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 80 (2000), 31–49.

[HB] D. R. Heath-Brown, Fractional moments of the Riemann zeta function, J. London Math. Soc. (2) 24 (1981), 65–78.

[IK04] H. Iwaniec, E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications 53, 2004.

[Odl87] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), 273–308.

[Mon73] H. L. Montgomery, The pair correlation of zeros of the zeta function, in Analytic Number Theory (Proc. Sympos. Pure Math., Vol.\ XXIV), American Mathematical Society, 1973, pp.\ 181–193.

[Odl] A. M. Odlyzko, Tables of zeros of the Riemann zeta function, https://www-users.cse.umn.edu/~odlyzko/zeta_tables/.

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Manuscript version 2.2 (April 2026). Corresponding author: Tamás Nagy (tnagyphd@gmail.com).